As the generation of the FTN signaling follows the convolutional structure, which can be interpreted by a trellis [1]}.
For the detection of FTN signaling, we define an error event as a segment of a path on the FTN trellis that diverges/remerges from/to the correct path [2]}.
For an FTN system with a block length of \(N\) , multiple error events can happen.
Specifically, suppose that the output of a correct path on the FTN trellis for a legitimate transmitted sequence is denoted by \(\mathbf {x}\)  and the output of the erroneous path is another legitimate transmitted sequence denoted by \(\mathbf {x}^{\prime }\) .
Then, for BPSK modulation, we define an error sequence as \(\mathbf {e} = \mathbf {x} - \mathbf {x}^{\prime }\) , where \(\mathbf {e} \in \mathcal {S}^{(N \times 1)}\)  and \(\mathcal {S}=\lbrace 0,+2, -2\rbrace \) .
According to the derivations in [3]}, [4]}, we find the following definitions and lemma are useful.
